[Complex Catenation]: What exactly is going on with the verbs in these sentences?

Allow takes part in certain single-object constructions:

They allow talking in the library.

But it is usually ditransitive (taking two objects):

They allow you one phone call.

She allowed him one meal a day.

It is also used in complex catenations with a to-infinitive:

They allowed him to go.

The ditransitive constructions are probably reduced forms of catenations:

She allowed him [to have] one meal a day.

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Let (the sense meaning permit) is used mainly in complex catenations with a bare infinitive:

They let him go/stay/speak/buy a bottle of wine.

She let him have one meal a day.

It is sometimes apparently monotransitive, as the bare infinitive may be dropped in informal contexts:

Did they let him? [verb understood from previous context]

But it is never used ditransitively.

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Permit is used mainly in complex catenations with a to-infinitive:

They permitted him to stay.

Though it is used sometimes ditransitively:

She permitted him one kiss.

And sometimes monotransitively:

We do not permit smoking in the dormitory.

It seems to me that allowed and let are not syntactically synonyms. Similar to watch and look. While similar concepts. The sentences "I watched TV." and "I looked at TV." are both correct, but we had to add the at.

Allowed seems like it implies a quantity already, where let is more like permitted. It's missing something, "She permitted him one meal a day." makes only slightly more sense than "She let him one meal a day.". Permitting and letting both don't specify what action is permitted, and so require another verb. Allowed gets away with this because it's like saying "let have" or "permitted to have" (while the sentence will sometimes need to be reorganized, like "permitted him to have"). The have concept is already included?

Allowed is acceptable in this context as it indicates that she 'allowed him to have' and there can be no confusion about what it means.

However, she let could cause some confusion, as it could possibly mean that she rented him a meal. 'To let' is also used to mean to rent out: http://dictionary.cambridge.org/dictionary/british/let_3